Hu-Washizu Variational Principle Research Articles

Variational Principles for Piezoelectric, Thermopiezoelectric, and Hygrothermopiezoelectric Continua Revisited

In this paper some variational principles are revisited for the fundamental equations of a regular region of piezoelectric, thermopiezoelectric, and hygrothermopiezoelectric (but non-stochastic, non-local, and non-relativistic) materials in the elastic range. Certain oversights, and especially, those involving the so-called Hu-Washizu variational principle of piezoelectricity that was first formulated in a paper (“Variational Principles in Piezoelectricity,” Lettere Al Nuovo Cimentò, vol. 7, 449–454, 1973) are clarified within the “ISI-Web of Science” publications in the open literature. Similar variational principles of piezoelectricity are cited.

A geometrically nonlinear thick plate bending element based on mixed formulation and discrete collocation constraints

In recent years there are many plate bending elements that emerged for solving both thin and thick plates. The main features of these elements are that they are based on mix formulation interpolation with discrete collocation constraints. These elements passed the patch test for mix formulation and performed well for linear analysis of thin and thick plates. In this paper a member of this family of elements, namely, the Discrete Reissner-Mindlin (DRM) is further extended and developed to analyze both thin and thick plates with geometric nonlinearity. The Von K<TEX>$\acute{a}$</TEX>rm<TEX>$\acute{a}$</TEX>n's large displacement plate theory based on Lagrangian coordinate system is used. The Hu-Washizu variational principle is employed to formulate the stiffness matrix of the geometrically Nonlinear Discrete Reissner-Mindlin (NDRM). An iterative-incremental procedure is implemented to solve the nonlinear equations. The element is then tested for plates with simply supported and clamped edges under uniformly distributed transverse loads. The results obtained using the geometrically NDRM element is then compared with the results of available analytical solutions. It has been observed that the NDRM results agreed well with the analytical solutions results. Therefore, it is concluded that the NDRM element is both reliable and efficient in analyzing thin and thick plates with geometric non-linearity.

Development of a discrete Reissner–Mindlin element on Winkler foundation

Over the years several plate-bending elements have been developed by several researchers. Most of the developed elements have been either for thin plates or thick plates. Recently, a set of robust triangular and quadrilateral elements, which showed excellent performance for both thin and thick plates, had been developed. Among the series of the triangular elements developed is the Discrete Reissner–Mindlin (DRM) element. The main features of the DRM element are its mixed interpolation formulation and its discrete collocation constraints. In this paper the DRM element has been extended to solve problems of plates on elastic foundations. The Hu–Washizu variational principle is used to formulate the element stiffness matrix. Deflection, bending moment and shear force responses of plates on Winkler-type foundation and with various thicknesses, loading conditions and foundation elastic constants, using both the DRM finite element and the classical approach, are presented. It has been demonstrated that the DRM finite element solutions agree very well with the classical solutions, specifically, for deflection response in the case of both thin and thick plates. Therefore, it can be concluded that, the DRM plate element is both effective and reliable for solving problems of thin and thick plates on elastic foundations.

A robust non-linear solid shell element based on a mixed variational formulation

The paper is concerned with a geometrically non-linear solid shell finite element formulation, which is based on the Hu-Washizu variational principle. For the approximation of the independent displacement, stress and strain fields, the strain field is additively decomposed into two parts. Due to the fact that one part of the strain field is interpolated in the same manner as proposed by the enhanced assumed strain (EAS) method, it is denoted as EAS field. The other strain field is approximated with the same interpolation functions as the stress field. In contrast to the EAS concept the approximation spaces of the stresses and the enhanced assumed strains are not orthogonal. Consequently the stress field is not eliminated from the finite element equations. For the displacements tri-linear shape functions are considered. Shear locking and curvature thickness locking are treated using assumed natural strain interpolations. A static condensation leads to a simple low order hexahedral solid shell element. Numerical tests show that the present model is very robust and allows larger load steps than an EAS solid shell element.

A URI 4-Node quadrilateral element by assumed strain

In this paper one-point quadrature “assumed strain” mixed element formulation based on the Hu-Washizu variational principle is presented. Special care is taken to avoid hourglass modes and volumetric locking as well as shear locking. The assumed strain fields are constructed so that those portions of the fields which lead to volumetric and shear locking phenomena are eliminated by projection, while the implementation of the proposed URI scheme is straightforward to suppress hourglass modes. In order to treat geometric nonlinearities simply and efficiently, a corotational coordinate system is used. Several numerical examples are given to demonstrate the performance of the suggested formulation, including nonlinear static/dynamic mechanical problems.

Non-conventional non-linear two-node hybrid stress-strain curved beam elements

This paper presents a family of two-node hybrid stress–strain curved beam elements with four displacement degrees of freedom (dof) per node for the finite deformation 2D Timoshenko beam theory, which can be readily generalized on the two-node curved beam elements with six displacement dof for the 3D beam theory. The developed formulation is based on the principally new non-linear strain–displacement relationships that are objective, i.e., invariant under arbitrarily large rigid-body motions. To avoid shear and membrane locking and have no spurious zero energy modes, the assumed stress resultant and strain fields are invoked. In order to circumvent thickness locking, the modified material stiffness matrices corresponding to the plane stress state are employed. The fundamental unknowns consist of four displacements and five strains of the face lines of the beam, and five stress resultants. The element characteristic arrays are obtained by using the Hu-Washizu variational principle. To demonstrate the efficiency and accuracy of this formulation and to compare its performance with other non-linear finite element models reported in the literature, extensive numerical studies are presented.

Finite element formulation for modelling large deformations in elasto‐viscoplastic polycrystals

AbstractAnisotropic, elasto‐viscoplastic behaviour in polycrystalline materials is modelled using a new, updated Lagrangian formulation based on a three‐field form of the Hu‐Washizu variational principle to create a stable finite element method in the context of nearly incompressible behaviour. The meso‐scale is characterized by a representative volume element, which contains grains governed by single crystal behaviour. A new, fully implicit, two‐level, backward Euler integration scheme together with an efficient finite element formulation, including consistent linearization, is presented. The proposed finite element model is capable of predicting non‐homogeneous meso‐fields, which, for example, may impact subsequent recrystallization. Finally, simple deformations involving an aluminium alloy are considered in order to demonstrate the algorithm. Copyright © 2004 John Wiley &amp; Sons, Ltd.

A mixed 8-node hexahedral element based on the Hu-Washizu principle and the field extrapolation technique

A mixed eight-node hexahedral element formulated via the Hu-Washizu principle as well as the field extrapolation technique is presented. The mixed element with only three translational degrees of freedom at each node can provide extremely accurate and reliable performance for popular benchmark problems such as spacial beams, plates, shells as well as general three-dimensional elasticity problems. Numerical calculations also show that when extremely skewed and coarse meshes and nearly incompressible materials are used, the proposed mixed element can still possess excellent behaviour. The mixed formulation starts with introduction of a parallelepiped domain associated with the given general eight-node hexahedral element. Then, the assumed strain field at the nodal level is constructed via the Hu-Washizu variational principle for that associated parallelepiped domain. Finally, the assumed strain field at the nodal level of the given hexahedral element is established by using the field extrapolation technique, and then by using the trilinear shape functions the assumed strain field of the whole element domain is obtained. All matrices involved in establishing the element stiffness matrix can be evaluated analytically and expressed explicitly; however, a 24 by 24 matrix has to be inverted to construct the displacement extrapolation matrix. The proposed hexahedral element satisfies the patch test as long as the element with a shape of parallelepiped.

Non-conventional non-linear two-node hybrid stress-strain curved beam elements

This paper presents a family of two-node hybrid stress–strain curved beam elements with four displacement degrees of freedom (dof) per node for the finite deformation 2D Timoshenko beam theory, which can be readily generalized on the two-node curved beam elements with six displacement dof for the 3D beam theory. The developed formulation is based on the principally new non-linear strain–displacement relationships that are objective, i.e., invariant under arbitrarily large rigid-body motions. To avoid shear and membrane locking and have no spurious zero energy modes, the assumed stress resultant and strain fields are invoked. In order to circumvent thickness locking, the modified material stiffness matrices corresponding to the plane stress state are employed. The fundamental unknowns consist of four displacements and five strains of the face lines of the beam, and five stress resultants. The element characteristic arrays are obtained by using the Hu-Washizu variational principle. To demonstrate the efficiency and accuracy of this formulation and to compare its performance with other non-linear finite element models reported in the literature, extensive numerical studies are presented.

Optimal solid shells for non-linear analyses of multilayer composites. II. Dynamics

We are presenting a simple low-order solid-shell element formulation––having only displacement degrees of freedom (dofs), i.e., without rotational dofs––that has an optimal number of parameters to pass the patch tests, and thus allows for efficient and accurate analyses of large deformable multilayer shell structures using elements at extremely high aspect ratio. With the dynamics referred to a fixed inertial frame, the elements can be used to analyze multilayer shell structures undergoing large overall motion. The formulation of this element is based on the mixed Hu-Washizu variational principle leading to a novel enhancing strain tensor (enhanced assumed strain (EAS) method) that renders the computation particularly efficient, with improved in-plane and out-of-plane bending behavior (Poisson thickness locking), especially in refined analyses of composite structures involving a large number of high aspect-ratio layers. The energy–momentum conserving algorithm in the context of current solid shell element is presented. We discuss the EAS formulation based on the displacement gradient and its complexity compared to formulation on the Green–Lagrange strain. Shear locking and curvature thickness locking are treated using the assumed natural strain (ANS) method. The element has an optimal combination of the ANS method and the minimal number of EAS parameters required to pass the plate bending patch test. Numerical examples involving dynamic analyses (with conservation of energy and momentum) of multilayer shell structures having a large range of element aspect ratios are presented. Several implicit direct integration methods with/without numerical dissipation are used and compared in terms of the accuracy, stability and cost in multilayer shell structure. Finally, we note that the topic in this paper is a fitting dedication to Professor Ekkehard Ramm, who has made important pioneering contributions in this research direction.

A 3D brick element based on Hu–Washizu variational principle for mesh distortion

AbstractIn order to improve the accuracy of the mixed element for irregular meshes, a penalty‐equilibrating 3D‐mixed element based on the Hu–Washizu variational principle has been proposed in this paper. The key idea in this work is to introduce a penalty term into the Hu–Washizu three‐field functional, which can enforce the stress components to satisfy the equilibrium equations in a weak form. Compared with the classical hybrid and mixed elements, this technique can efficiently reduce the sensitivity of the element to mesh distortion. The reason for the better results of this penalty technique has been investigated by considering a simple 2D problem. From this investigation, it has been found that the penalty parameter here plays the role of a scaling factor to reduce the influence of the parasitic strain or stress, which is similar to the devised selective scaling factor proposed by Sze. Furthermore, compared with the hybrid stress element, the proposed element based on the three‐field variational principle is more suitable for material non‐linear analysis. Numerical examples have demonstrated the improved performance of the present element, especially in stress computation when FEM meshes are irregular. Copyright © 2002 John Wiley &amp; Sons, Ltd.

Well‐posedness and numerical performances of the strain gap method

AbstractA mixed method of approximation is discussed starting from a suitably modified expression of the Hu‐Washizu variational principle in which the independent fields are displacements, stresses and strain gaps defined as the difference between compatible strains and strain fields. The well‐posedness of the discrete problem is discussed and necessary and sufficient conditions are provided. The analysis of the mixed method reveals that the discrete problem can be split into a reduced problem and in a stress recovery. Accordingly, the discrete stress solution is univocally determined once an interpolating stress subspace is chosen. The enhanced assumed strain method by Simo and Rifai is based on an orthogonality condition between stresses and enhanced strains and coincides with the reduced problem. It is shown that the mixed method is stable and converges. Computational issues in the context of the finite element method are discussed in detail and numerical performances and comparisons are carried out. Copyright © 2001 John Wiley &amp; Sons, Ltd.

A variationally consistent generalized variable formulation for enhanced strain finite elements

The so-called enhanced strain finite elements are based on the enrichment of the standard compatible strain field by the introduction of additional, non-compatible strains. This class of elements can be derived starting from a partial Hu–Washizu variational principle. However, since in the original enhanced strain formulation the stress field is eliminated from the formulation, a separate least-squares procedure had to be implemented for a variational derivation of the stress field. A three-field generalized variable approach incorporating strain enhancement is proposed in the present paper within the context of linear elastic structural problems. It is shown how the original two-field enhanced strain method can be naturally recovered by suitably choosing the strain model. For this case a straightforward, but still variationally consistent stress recovery is proposed. Copyright © 2000 John Wiley & Sons, Ltd.

A variationally consistent generalized variable formulation for enhanced strain finite elements

The so-called enhanced strain finite elements are based on the enrichment of the standard compatible strain field by the introduction of additional, non-compatible strains. This class of elements can be derived starting from a partial Hu–Washizu variational principle. However, since in the original enhanced strain formulation the stress field is eliminated from the formulation, a separate least-squares procedure had to be implemented for a variational derivation of the stress field. A three-field generalized variable approach incorporating strain enhancement is proposed in the present paper within the context of linear elastic structural problems. It is shown how the original two-field enhanced strain method can be naturally recovered by suitably choosing the strain model. For this case a straightforward, but still variationally consistent stress recovery is proposed. Copyright © 2000 John Wiley & Sons, Ltd.

Further study of the equivalent theorem of Hellinger-Reissner and Hu-Washizu variational principles

In this paper, it has been proved that the well-known Hu-Washizu variational principle is a pseudo-generalized variational principle (pseudo-GVP). The stationary conditions of its functional may satisfy all its field equations and boundary conditions if all the variables in the functional are considered as independent variations, but there might exist some kinds of constraints. Some new pseudo-GVPs are established to distinguish them from genuine ones by the so-called inverse Lagrange multiplier method. The constrained Hu-Washizu principle, therefore, is proved to be equivalent with the Hellinge-Reissner principle under the constraints of stress-strain relations.

Analytical response sensitivity computation using hybrid finite elements

This paper presents analytical sensitivity computation using hybrid finite elements. Expressions have been derived for analytical response sensitivities using two-dimensional hybrid elements developed on the basis of the modified Hu–Washizu variational principle. Computational algorithms have been formulated for sensitivity computation and the same have been implemented using a partial symbolic computational scheme, verification being done with standard bench-mark examples. Illustrations of the response sensitivity computation with respect to sizing variables have been presented. The improvement in the accuracy of sensitivity values using hybrid elements as compared with conventional displacement based elements is examined.

Equivalent theorem of Hellinger-Reissner and Hu-Washizu variational principles

The paper has proved that Hellinger-Reissner and Hu-Washizu variational principles are but equivalent principles in elasticity by following three ways: 1) Lagrange multiplier method. The paper points out that only a new independent variable can be introduced when one constraint equation has been eliminated by one Lagrange multiplier, which must be expressed as a function of the original variable(s) and/or the new introduced variable after identification. In using Lagrange multiplier method to deduce Hu-Washizu principle from the minimum potential energy principle, which has only one kind of independent variable namely displacement, by eliminating the constraint equations of stress-displacement relations, one can only obtain a principle with two kinds of variables namely displacement and stress; 2) involutory transformation, with such method Hu-Washizu variational principle can be deduce directly from the Hellinger-Reissner variational principle under the same variational constraints of stress-strain relation, and vice verse; 3)semi-inverse method, by which both of the above variational principles can be deduced from the minimum potential energy principle with the same variational constraints. So the three kinds of variational functions in Hu-Washizu variational principle are not independent to each other, the stress-strain relationships are still its constraint conditions.

Thermo-Viscoplastic Modeling of Machining Process Using a Mixed Finite Element Method

This paper considers a thermo-viscoplastic model of the steady state orthogonal machining process by using a three-field mixed finite element method based on the Hu-Washizu variational principle. Assuming a trial chip geometry, work and tool material properties, and cutting conditions (cutting speed, feed rate and rake angle), detailed information on the state of the stresses, deformation, and temperature distributions in the workpiece and tool are obtained. This approach is shown to satisfy the nontrivial stress boundary condition better than the machining model that is based on the compatible displacement finite element method. The heat generated due to the plastic deformation, to the friction, and the heat conducted into the tool are calculated. The effects of the heat conduction and friction on the temperature field are studied. The isothermal machining case is also considered to study the thermal effect on the machining process. Numerical results are given for the aluminum 6061 alloy and pure titanium under several cutting conditions.

Unified and mixed formulation of the 8-node hexahedral elements by assumed strain method

In this paper, a class of ‘assumed strain’ mixed finite element methods based on the Hu-Washizu variational principle is presented. Special care is taken to avoid hourglass modes and shear locking as well as volumetric locking. A unified framework for the 8-node hexahedral solid and thermal as well as thermomechanical coupling elements with uniform reduced integration (URI) and selective numerical integration (SI) schemes is developed. The approach is simply implemented by a small change of the standard technique and is applicable to arbitrary non-linear constitutive laws including isotropic and anisotropic material behaviors. The implementation of the proposed SI elements is straightforward, while the development of the proposed URI elements requires ‘anti-hourglass stresses’ which are evaluated by classical constitutive equations. Several numerical examples are given to demonstrate the performance of the suggested formulation, including mechanical problems, heat conduction and thermomechanical problems with emphasis on sheet forming processes.

New stress recovery formula for B‐bar methods

AbstractAn alternative justification of the B‐bar procedure using the Hu–Washizu variational principle is presented. In contrast to previous work, the derivation starts from the three groups of algebraic equations (strain‐displacement equations, stress‐strain equations and equations of equilibrium) obtained by discretization of the Hu–Washizu functional. Based on an appropriate assumption, these equations can be manipulated to yield the element stiffness matrix in the form typical of the B‐bar procedure. This approach directly leads to a stress‐recovery formula different from the one suggested before. The theory is illustrated by application to a family of quadrilateral plane elements.